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**Finite approximations to the critical reversible nearest particle system.**
*(English)*
Zbl 0966.82013

Summary: Approximating a critical attractive reversible nearest particle system on a finite set from above is not difficult, but approximating it from below is less trivial, as the empty configuration is invariant. We develop a finite state Markov chain that deals with this issue. The rate of convergence for this chain is discovered through a mixing inequality in [M. Jerrum and A. Sinclair, Inf. Comput. 82, No. 1, 93-133 (1989; Zbl 0668.05060)]; an application of that spectral gap bound in this case requires the use of “randomized paths from state to state.” For applications, we prove distributional results for semiinfinite and infinite critical RNPS.

### MSC:

82C22 | Interacting particle systems in time-dependent statistical mechanics |

60K35 | Interacting random processes; statistical mechanics type models; percolation theory |

### Keywords:

spectral gap; critical attractive reversible nearest particle system; finite state Markov chain; rate of convergence; mixing inequality### Citations:

Zbl 0668.05060
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\textit{T. Mountford} and \textit{T. Sweet}, Ann. Probab. 26, No. 4, 1751--1780 (1998; Zbl 0966.82013)

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### References:

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[12] | SWEET, T. 1997. One dimensional spin systems. Ph.D. thesis, Univ. California, Los Angeles. |

[13] | LOS ANGELES, CALIFORNIA 90095-1555 E-MAIL: malloy@math.ucla.edu |

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